Measuring Risk and Return

Chapter 5

2

Learning Objectives Outline the key factors that influence interest

rates

Describe the Fisher effect and its influence on interest rates and inflation

Calculate Risk and Return Measures

Discuss the characteristics of the Normal distribution

Understand the historic returns on risky portfolios

3

Rate of Return Basic Formula

Keep these in mind ALL SEMESTERYou will need these throughout the whole semester

Single Period Rate of Return Perpetuity Growing Perpetuity

4

Single Period Rates of Return

Invested

Change

P

DCG

P

DPPHPR$

$

0

11

0

101

P0 = Beginning price

P1 = Ending price

D1 = Dividend during period one

CG1 = Capital Gains during period one

Holding Period Return (single period):

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HPR Break Down

We earn returns in one of two ways Capital Gains: Price Appreciation

Capital Gains Yield is CG1/P0

DividendsDividend Yield is D1/P0

P

D

P

CG

P

DCG

P

DPPHPR0

1

0

1

0

11

0

101

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Holding Period Return Questions The current price of a stock is $25, you expect

the stock’s price at the end of the period to be $29.50 after paying a dividend of $0.50. What is the holding period return?

What is the capital gain yield?

What is the dividend yield?

7

Remember Stocks are Perpetuities

What is the price of a stock with a $10 dividend if the discount rate is 10%?What is the price next year?

What is the price of a stock with a $10 dividend next year if grow is 2% and the discount rate is 10%?What is the price next year?

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Expected Return HPR tells us what we earned, but investments

are based on what we EXPECT to earn Expectations are based on the possible states of

the worldr(s) outcome if state occursp(s) probability that the state occurs

Expected return is the weighted average of the possible returns

( ) ( ) ( )s

E r p s r s

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State Prob. of State HPR in State Excellent .25 0.3100Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200

Scenario Analysis: Possible States of Nature and Holding Period Returns

In this set-up there are only 4 possible “states of nature” (investment outcomes).

Each “state” is associated with: a probability of that state occurs, and the return on the investment if the state occurs

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Backwards Inducing Price

We expect an investment is equally likely to payoff either $125,000; $75,000; or -$20,000 next year. If we demand a return of 20%, how much are we willing to pay?

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Nominal v Real

Nominal Dollar:The dollar in your wallet, or bank account

Real Dollar:Refers to purchasing power

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Nominal v Real Dollar Example Hershey Nickel Bar Example

In 1930 bar was 2 ozIn 1968 bar was ¾ oz

How much does it cost (nominal & real) to buy 2 oz of chocolate?

1930 1968

Real Nickel buys 2oz 2oz

Nominal Nickel buys 2oz 0.75oz

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Real and Nominal Rates of Return

Nominal interest rate (“rn”)Growth rate of your money

Real interest rate (“rr”)Growth rate of your purchasing power

Inflation rate (“i”): The general decline in what a dollar can purchase

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Taco World Tacos are the only good in our world

Cost: $1/Taco We can invest $100 today and earn 20%

Forgo tacos today for more tacos next year How many tacos can we buy next year?

Nominal HPR? Real HPR?

Year 0 Year 1

Invest @ 20% $100 $120

Taco Price $1/taco $1/taco

Tacos 100 Tacos 120 Tacos

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Taco World Tacos are the only good in our world

Cost $1/Taco: Inflation is 9.1% We can invest $100 today and earn 20%

Forgo tacos today for more tacos next year How many tacos can we buy next year?

Nominal HPR? Real HPR?

Year 0 Year 1

Invest @ 20% $100 $120

Taco Price $1/taco $1.091/taco

Tacos 100 Tacos 110 Tacos

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Equilibrium Nominal Rate of Interest

As the inflation rate increases, investors will demand higher nominal rates of return

If E(i) denotes current expectations of inflation, then we get the Fisher Equation:

(1+rn) = (1+rr) * (1+E(i))

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Real vs. Nominal Rate Example

If you invest $10,000 at a nominal rate of 12% APR, how much will you have in 30 years?

How much will you have in real terms if the rate of inflation is 4% per year?

What is your nominal RoR? Real RoR?

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Equilibrium Real Rate of Interest

Real Rate Determined by:Supply

Household savings

Demand Business Investment

Government actions Federal Reserve

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Determination of the Equilibrium Real Rate of Interest

Government Increases Deficit

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Bills and Inflation, 1926-2012

Moderate inflation can offset most of the nominal gains on low-risk investments.

A dollar invested in T-bills from 1926–2012 grew to $20.25, but with a real value of only $1.55.

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Risk & Risk Premium

If T-Bills and Google both have an expected return of 10%, where does the average person invest?

Why?

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Risk and Risk Premiums

Risk Aversion: People generally dislike risk To induce people to take on risk they must be

rewarded with higher returnsRisk Premium: Difference between the expected

RoR and the risk-free rateExcess Return: Difference between the actual RoR

and the risk-free rate

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Measuring Risk What is risk? There is no universally agreed-upon

measureHowever, variance and standard deviation are both

widely accepted measures of total risk

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Statistics Review: Variance Variance (σ2) measures the dispersion of

possible outcomes around expected return Standard deviation (σ) is the square root of

variance Higher variance (std dev), implies a higher

dispersion of possible outcomesMore uncertainty

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Different Variances

Possible ReturnsE(r)

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Variance (VAR):

Variance and Standard Deviation

22 ( ) ( ) ( )s

p s r s E r

2STD

Standard Deviation (STD):

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Example You invest in a stock at the current price of

$50. Your expectation regarding the price and the dividend in the following year depends upon how the economy performs:

Compute the expected return and standard deviation of this investment

Economy Probability Dividend Ending Price

Strong 30% $2.00 $60.00

Normal 50% $1.00 $54.00

Weak 20% $0.50 $44.50

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Using the Time Series of Historical Returns We cannot determine the “true” mean and

variance of an investment because we don’t know all the possible scenarios

Therefore we often estimate the mean and variance based on historical information

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Using Historical Returns Each observation is a “scenario” We view each is equally likely of recurring

If there are “n” observations then each scenario’s probability of occurring is 1/n

The expected return is:

Where p(s) = 1/n

n

ssrsprE

1)()()(

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When E(r) is less informative

Does an investor really expect to earn a 1 year HPR of 0%?(0.25*0.2)+(0.25*-0.2)+(0.25*0.2)+(0.25*-0.2)

1996 1997 1998 1999

20% -20% 20% -20%

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Geometric Average Return Used to compute the average compound return

of an investment over multiple periods

rg= geometric average rate of return

1)]1(*..*)1(*)1[(1

21 nnrrrrg

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When E(r) is less informative

Arithmetic Average is 0%?(0.25*0.2)+(0.25*-0.2)+(0.25*0.2)+(0.25*-0.2)

Geometric Average is:

1996 1997 1998 1999

20% -20% 20% -20%

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Arithmetic v Geometric Average RoR

The arithmetic average rate of return answers the question, “what was the average of the yearly rates of return?”

The geometric average rate of return answers the question, “What was the growth rate of your investment?”

34

More on Average Returns The geometric average will be less than the arithmetic

average unless all the returns are equal. Arithmetic average is an overly optimistic estimate of

future returns for long horizons. The geometric average is an overly pessimistic

estimate of future returns for short horizons.

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Forecasting Return

To achieve a more accurate estimate of expected returns, one can use Blume’s formula:

AverageArithmeticN

TNverageGeometricA

N

TTR

11

1)(

where, T is the forecast horizon and N is the number of years of historical data we are working with. T must be less than N.

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Blume Example

Over a 30-year period a stock had an arithmetic average of 15% and a geometric average of 11%. Using Blume’s formula what is the best estimate of the future annual returns over the next 5 years? 10 years?

AverageArithmeticN

TNverageGeometricA

N

TTR

11

1)(

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Dollar Weighted Average Return

The internal rate of return earned on an investment

Gives an idea of what the investor actually earned

Treat the investment like a corp capital budgeting problem and find the IRR

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Dollar Weighted Example 2008 bought 100 shares @ $50 2009 return 10% buy another 50 shares, $2 div/sh 2010 shares sold 75 @ $51, $4 div/sh 2011 shares sold 75 @ $54 $4.5 div/sh What are the cash flows?

Year Price Share Cash Flow2008 50 1002009 502010 51 752011 54 75

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Example Continues

Use the CF button to find the IRR 2nd CE/C CF – Cashflow – Enter – Down Arrow Fill in all Cashflows IRR - CPT

Year Price Share Cash Flow2008 50 100 -5,000.002009 55 50 -2,650.002010 51 75 4,125.002011 54 75 4,387.50

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Historical Risk Estimated Variance = expected value of squared

deviations

Estimated Variance is biased downward We are using the historical average instead of the actual

expected return To eliminate the bias we modify the variance formula

2

1

_2 1

ˆ

n

s

rsrn

2

1

_2

1

1ˆ

n

j

rsrn

41

Historical Risk Example

What is the historical variance of the Index?The average return over the period is 6.4%

2010 2011 2012 2013 2014

8 9 5 4 6

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Comparing Investments

Investment A earned 20% Investment B earned 8% Which did better?

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Comparing Investments

We cannot simply compare returns when we compare investments. WHY?FYI: How are mutual funds advertised?

To fairly compare investments we need to examine both the return earned and the risk involved → Sharpe Ratio

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Sharpe Ratio: Reward to Volatility A measure of risk adjusted performance

Is higher return due to good performance or more risk?

Higher Sharpe Ratios → a more efficient investmentA better risk return trade off

h𝑆 𝑎𝑟𝑝𝑒𝑅𝑎𝑡𝑖𝑜=𝑅𝑖𝑠𝑘𝑝𝑟𝑒𝑚𝑖𝑢𝑚

𝑆𝑡𝑑 𝐷𝑒𝑣𝑜𝑓 𝑒𝑥𝑐𝑒𝑠𝑠𝑟𝑒𝑡𝑢𝑟𝑛𝑠

45

Comparing Investments

Higher Return due to performanceInvestment A earned 20% - SR 3Investment B earned 8% - SR 1

Higher Return due to riskInvestment A earned 20% - SR 1Investment B earned 8% - SR 3

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Risk Return TradeoffR

etur

ns

Risk

T-Bills

T-Bonds

LT Corp Bonds

Large Cap Stock

ST Corp Bonds

Small Cap Stock

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Given $100,000 to invest: What is the expected return and standard deviation

of each of the investment opportunities?

What is the expected risk premium in dollars of investing in equities versus risk-free T-bills?

Invest in: Probability Return ($)

Equities .6 $50,000

.4 -$30,000

T-Bills 1.0 $5,000

48

Annualizing Returns Annual Percentage Rate (APR): This is the return

commonly discussedCredit Cards, LoansFound using simple interest

Effective Annual Rate (EAR): Return an investment actually makes over a yearFound using compound interest

EAR = {1+ (APR/n)}n – 1n is the number of compounding periods per year

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Examples: APR & EAR What is the EAR of a 4% APR that compounds:

Semi-annually?

Quarterly?

Monthly?

What is the APR and EAR of 0.5% perWeek

Month

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High Math Investment Foundation Underlying most of the class is the idea that

returns are normally distributedThis assumption is central to investment theory

and practice Implications

If security returns are normal then so are portfolioStandard deviation and the mean completely

describe the distributionStandard deviation is an appropriate measure of

risk Fortunately, returns appear to normal

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The Normal Distribution

Mean = 10%, SD = 20%

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Normal Distribution Example

A security with normally distributed returns has an annual expected return of 18% and standard deviation of 23%. What is the probability of getting a return between -28% and 64% in any one year?

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Measuring “Surprise”

Standard Deviation Score: How much of a surprise an observed returnsr i = [r i – E(r i)] / σi

Return surprise divided by standard deviation

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Historical Distribution of Monthly Returns

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How Much Could I loss?

Value at Risk (VaR): What is the worse loss that an investment will suffer, given a probability (often 5%)

VaR = E(r ) + (-1.64485* σ)VaR at 5% with normal return distribution

What is my value at risk on an investment with an expected return on 12%, and a standard deviation of 5%

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Non-Normal Distributions

When distributions are non-normal we need to consider more than mean and variance

57

Distribution Characteristics Mean

Most likely outcome Variance or standard deviation

The spread of possible outcomes Skewness

How asymmetrical is the distribution Kurtosis

Flat or “Peakie”* If a distribution is approximately normal, the distribution is

described by the mean and standard deviation

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Normal and Skewed Distributions

Mean = 6%, SD = 17%

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Normality and Risk Measures

What if excess returns are not normally distributed?Standard deviation is no longer a complete

measure of riskSharpe ratio is not a complete measure of

portfolio performanceNeed to consider skewness and kurtosis

60

61

Expected Return Example Amy has just purchased 1,000 shares of GE.

She expects that the return over the next year will depend on the state of the economy. Given her expectations what is her expected return?

State Probability Expected Return

Boom 10% 35%

Normal 70% 12%

Bust 20% -18%

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Variance and Standard DeviationState Prob. of State r in State Excellent .25 0.3100Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200E(r) = 9.76%Variance = ?Standard Deviation = ?

63

Historical Risk Example

Expected RoR

Variance =

Standard Deviation =

1996 1997 1998 1999 2000

20% 15% -5% 5% 10%